981255
An Experimentally Validated Nonlinear Stabilizing Control for
Power Electronics Based Power Systems
Steven F. Glover
P.C. Krause and Associates
Scott D. Sudhoff
Purdue University
Copyright © 1997 Society of Automotive Engineers, Inc.
ABSTRACT
High performance high bandwidth control of power
electronic converters, inverters, and motor drives has
become feasible over the past decade. These devices
behave as constant power loads over large bandwidths
when they are tightly regulated. However, constant
power loads have a severe side affect known as
negative impedance instability. In order to mitigate the
problem of negative impedance instability a new
nonlinear system stabilizing controller has been
developed. The details of how this controller works
along with its implementation is discussed and
demonstrated in hardware.
INTRODUCTION
Power electronics based power and propulsion
systems are topics of great interest to industry and the
military. Applications include submarines, ships, hybrid
electric and electric vehicles, aircraft, and spacecraft.
The flexibility and potential for high bandwidth control in
these systems, which is afforded by the use of power
electronic converters/inverters, reduces the need for
human operators and at the same time permits a greater
degree of optimization, reducing fuel costs. As a result
of fuel savings, these systems also offer lower
emissions. However, with these benefits come one
disadvantage in that these systems tend to be inherently
dynamically unstable.
In such a system, most loads take the form of electric
drive systems, dc/ac converters, or dc/dc converters. In
the past, many different control schemes have been
investigated including Sliding Mode Control [2], Fuzzy
Logic [3 and 4], Nonlinear Proportional Integral (PI) [5],
as well as many others linear and nonlinear [6 and 7].
Regardless of the strategy, these different control laws
share the property that if they tightly regulate the
converter/inverter output, the converter/inverter presents
a constant power load to the system. Thus, the current
entering the converter/inverter increases if the voltage at
the input decreases, causing the converter/inverter to
display a negative impedance characteristic as viewed
from the rest of the system - clearly a destabilizing
effect.
This has led to an impedance based stability criteria
that predicts, on an operating point basis, when the
overall system will be unstable. Using this criteria there
are several techniques that can be used to eliminate the
negative impedance instability. The first of which is
designing the converter regulatory control with very low
bandwidth so that the negative impedance characteristic
is reduced. This can be effective but results in poor
regulation of the output of the converters/inverters.
Another approach is to incorporate a passive RC
damping network to stabilize the system. This is also
effective but the extra hardware and space required can
be quite expensive, and system efficiency is reduced.
The path chosen here is to modulate the commanded
input power of the load converters at the appropriate
frequencies with a nonlinear system stabilizing controller
(NSSC). One possibility to accomplish this is to monitor
the power passed down the AC transmission line [8] and
from this measurement modulate the commanded
converter/inverter input power. This requires additional
sensors
placed
long
distances
from
the
converter/inverter. Another possibility is the use of a
linear feed forward pole-zero cancellation control as
discussed in [9] to eliminate the instability. This linear
feed forward control requires a duty cycle controlled
converter/inverter with the assumption that the input
voltage times the on time of the upper transistor is a
constant. The technique chosen here, [10 and 11], uses
a new nonlinear feed forward control law that is not only
simple to implement but also has minimal effect on the
desired performance of the system and at the same time
guarantees system stability. This nonlinear control law
works well with current controlled converters/inverters,
which allow better control in terms of safety and current
limiting. The NSSC is augmented to the existing
controls which are designed solely on the basis of output
regulation. This new control scheme is verified on both
the induction motor drive (IMD) of a propulsion system,
and the dc/dc buck boost converter (BBC) of a DC
power system.
Figure 2 (note that the control proposed in this paper, is,
however, independent of whether or not the field
oriented control is direct or indirect). Therein, an
BASE TEST SYSTEMS
instantaneous torque command Te is the input to the
controller. This torque command is equal to the torque
desired by the controller governing the mechanical
dynamics, Te ,des . As can be seen, based on the torque
*
INDUCTION
MOTOR
BASED
ELECTRIC
PROPULSION SYSTEM - Figure 1 illustrates the type of
electric propulsion system considered herein, [1]. The
power source of the system is a diesel engine or turbine
(emulated by a dynamometer), which serves as a prime
mover for the 3-phase synchronous machine (SM). The
3-phase output of the machine is rectified using an
uncontrolled rectifier. The rectifier output voltage is
denoted v r . An LC circuit serves as a filter, and the
output of this filter is denoted vdcs . A voltage regulator /
exciter adjusts the field voltage of the SM in such a way
that the source bus voltage vdcs is equal to the
*
commanded bus voltage v dcs . The source bus is
connected via a tie line to the load bus, the voltage at
which is denoted vdci . The load bus consists of a
capacitive filter (which includes both electrolytic and
polypropylene capacitance) as well as a 3-phase fully
controlled inverter, which in turn supplies an induction
motor. The induction motor drives the mechanical load,
which is rotating at a speed ω rm,im . Based upon the
command Te and desired d-axis rotor flux level λdr , the
*
e*
e*
e*
desired q- and d- axis stator currents, i qs and i ds , are
determined.
This calculation is a function of the
induction motor rotor magnetizing inductance Lm , the
induction motor rotor inductance (rotor leakage plus
magnetizing) L' rr , the rotor resistance r' r , and the
number of poles. Based on the q- and d- axis stator
currents the electrical radian slip frequency, ω s ,im , is
determined, which is then added to the electrical rotor
speed ω r ,im in order to determine the electrical speed of
the synchronous reference frame Θ e,im . In addition to
the algorithm illustrated in Figure 2, especially in large
drives, the field oriented control will often include an on
line parameter identification algorithm to compensate for
variations of the rotor time constant [13-14].
mechanical rotor speed, and the desired electromagnetic
torque Te ,des (which is determined by the controller
governing the mechanical system), the induction motor
controls specify the on/off status of each of the inverter
semiconductors in such a way that the desired torque is
obtained. Although this system is quite robust with
regard to over currents, and simple to design from the
viewpoint that the controller governing the mechanical
system is decoupled from the control of the electrical
system (since the torque can be controlled nearly
instantaneously), such systems are prone to be subject
to a limit cycle behavior in the dc bus voltage known as
negative impedance instability [12].
Before setting forth the implementation of the
proposed NSSC controller, it is appropriate to first
consider a standard field oriented control such as the
rotor flux indirect field oriented control illustrated in
∑
Figure 2. Rotor Flux Oriented Indirect Field Oriented
Control.
Figure 1. System Configuration
Once the q- and d- axis current commands and the
position of the synchronous reference frame are
established, these currents may be synthesized in a
variety of ways. Herein, the q- and d- axis current
command was transformed back into an abc variable
current command, which is an input to a hysteresis type
current control.
System Behavior – The performance of the
propulsion system was tested by ramping the desired
electromagnetic torque of the induction motor from 2 to
19 Nm over a period of 100 ms. Figure 3 depicts the
*
commanded a-phase current i as , the actual a-phase
current ias , and the dc inverter voltage vdci . The
increase in torque can be associated with the linear
increase in ias . It can be seen that as the power
command increases the dc bus voltage becomes
unstable, stressing both the semiconductors and the
capacitors. In a typical system such behavior could
easily result in the semiconductor and/or capacitor
failure.
DC POWER SYSTEM - In order to investigate the
control of dc power systems, the small but
representative system depicted in Figure 4 was utilized,
[1]. As can be seen, this 3.7kW system consists of a
generation system, a distribution system, and loads.
The loads are the dc/dc converter, which is of special
concern herein, and a permanent magnet synchronous
motor drive.
Figure 3. Performance of standard field oriented control
during ramp increase in desired torque.
The generation system consists of a dynamometer
that acts as a prime mover, a 3-phase synchronous
machine, an LC filter, and a solid state exciter/voltage
regulator. The output of the rectifier is filtered by an LC
circuit creating a nearly ripple free dc source. The
generation system output is connected to a generation
bus, v dc1 , which is attached to the transmission line.
The transmission line transfers energy to the load bus,
v dc 2 , which distributes power to the remainder of the
system.
Vdc2
Vdc1
Figure 4. DC Power system
Vdc3
The dc system has two loads, a permanent magnet
synchronous machine drive (brushless dc motor) and a
BBC, Figure 5. The synchronous motor drive is used to
represent a propulsion load whereas the BBC represents
a distribution type of load in which a dc/dc power
converter is used to interface between two voltage
levels, and/or provide a voltage regulated bus from an
unregulated dc source. The operation and parameter
values of this motor drive are as set forth in [17] with the
exception that the current control was delta modulated at
a frequency of 30 kHz rather than hysteresis modulated
[18]. The modeling of this type of drive is set forth in
[17].
The nominal converter control scheme utilized
herein consists of two separate feedback levels. The
outer level consists of a regulating nonlinear PI
controller, Figure 6, used to maintain a constant output
voltage. It consists of second order low pass filters used
to eliminate aliasing in the measured inputs and to
remove discritization noise in the controller outputs. The
nonlinear block following the PI control converts the
output current command to an input current command.
System components are protected by limiting the range
of the commanded input current following the controller.
The conditional block is used to limit the valid operating
range of the converter based on the level of the
distribution bus voltage, providing additional system
protection. The regulating PI controller was designed
based on the linearized average value model of the
system [15], resulting in controller gains of Kp and Ki
being equal to 0 and 3.1 respectively.
The inner level consists of a hysteresis current
controller that regulates the input current of the converter
to with in plus or minus a given hysteresis level of the
commanded input current.
Advantages of using
hysteresis current control are that current ripple is
independent of operating conditions and the tight
regulation of the input current provides for highly
effective current limiting. The two main disadvantages in
using this type of control are variable switching
frequency and an undefined duty cycle, which makes the
average value model difficult to derive as can be seen by
the almost complete avoidance of this type of control in
literature. However, recently an appropriate average
value model has been set forth [15 and 16] and is the
approach used herein.
System Behavior – Figure 7 depicts the load
variables associated with the BBC for a step change in
BBC load from 129.3 to 60.1 Ohms. Depicted are the
output voltage of the BBC v dc3 , the distribution bus
500
vdc3 400
300
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
0
0
0.1
0.2
0.3
Time (sec)
0.4
0.5
400
vdc2 300
200
20
idcdc
10
Figure 5. Buck/Boost Converter Schematic
Figure 6. Nonlinear PI controller
Figure 7. Step in BBC load
voltage v dc 2 , and the current entering the BBC i dcdc . It
is shown that the system remains stable and recovers.
The current entering the BBC is shown limiting as set by
the controls but also recovers as v dc 2 approaches
steady state.
2
x 10
5
1.5
Feasible Region for
Pole Pair 1
1
0.5
Imag.
Axis
Figure 8 depicts the load variables for a study in
which the filter capacitance connected to the generation
bus is stepped from 1315.5 µF to 1.4 µF. Note how
v dc 2 and idcdc begin to oscillate violently once the
capacitance is removed, demonstrating that the original
control cannot operate without significant generation bus
capacitance. This is not surprising in view of the fact
that the control design assumed the presence of the
generation bus capacitance. This raises the question of
how much bandwidth would have to be sacrificed in
order to eliminate the generation bus capacitance from
the system with the standard PI control.
0
-0.5
-1
-1.5
-2
-1
-0.5
0
Real Axis
0.5
1
x 10
5
Figure 9. System Root Loci for Varying Kp and Ki
Feasible Region for
Pole Pair 2
(Conjugate Area Not Shown)
2000
1000
Feasible Region for
Pole Pair 3
(Conjugate Area Not Shown)
500
v dc3
400
300
400
v dc2
0
0.1
0.2
0.3
0.4
0.5
20
0
-1000
300
200
idcdc
Imag.
Axis
-2000
0
0.1
0.2
0.3
0.4
0.5
-2000
0
2000
4000
Real Axis
10
0
Figure 10. System Root Loci for Varying Kp and Ki
0
0.1
0.2
0.3
Time (sec)
0.4
0.5
Figure 8. Loss of Generation Bus Capacitance
As it turns out, the instability demonstrated in Figure
8 was found to be uncorrectable by adjustment of the
proportional integral control gains. In order to
demonstrate this, the system nonlinear average value
model (NLAM) was linearized and the loci of system
roots were plotted as the gains kp and ki were varied
between zero and nine million as shown in Figures 9 and
10. Once the generation bus capacitance is removed
there are three pole pairs which can contribute to system
instability. The shaded regions in Figure 9 (pole pair 1)
shows where the first pole pair can be placed while
Figure 10 (pole pairs 2 and 3) shows the areas in which
one pole of each of the remaining two interesting pole
pairs can be placed. Each pole pair can be made stable
but it is impossible to move all poles into the left hand
plane simultaneously. A similar situation can arise with
other converter configurations as well. This would seem
to mandate that at least some level of generation bus
capacitance must be present. However, this is not the
case. The NSSC algorithm introduced in the next
section can achieve system stability when it is
augmented to the regulating control regardless of the
generation bus capacitance without significantly
degrading the transient performance.
NONLINEAR SYSTEM STABILIZING CONTROLLER
The effect of regulating the constant power output of
the IMD and BBC results in the input of each device
appearing as a constant power, negative impedance
load.
Negative impedance loads in many power
systems have a destabilizing effect known as negative
impedance instability. The physical cause of, a measure
for prediction, and a means of mitigating this type of
instability are set forth in this section.
NEGATIVE IMPEDANCE INSTABILITY - In order to
gain insight into the nature of negative impedance
instability, consider the highly simplified representation
of both systems depicted in Figure 11. Therein, the
source is modeled as an ideal source followed by a low
pass filter; the constant power load could be considered
as either the IMD or the BBC. It is assumed that the
IMD as well as the BBC both compensate
instantaneously to changes in the dc link voltage, vin ,
allowing them to be modeled as single dependent
current sources which are formulated by assuming that
the input power is equal to the commanded input power,
P* .
Note that in a small signal sense the constant power
load appears as a negative resistance, which would
suggest a destabilizing effect. In terms of the linear
input impedance the stability criteria (3) may be
expressed
− Zin >
Leq
(6)
ReqCeq
As P increases − Z in decreases and so eventually the
system becomes unstable. Equation (6) immediately
suggests several methods for manipulating system
stability.
First, increasing the capacitance to an
appropriate level can insure stability. However, such
measures can be expensive in terms of capitol, space,
weight and reliability. Alternatively, reducing Leq is also
*
Figure 11. Simplified System Model
The stability of this simplified system can be
determined by calculating its pole locations.
The
differential equations governing the system can be
expressed as
pidceq =
v dceq − Reqidceq − v in
Leq
(1)
and
idceq − vP*
in
pvin =
C eq
(2)
P* <
Req Ceq vino 2
Leq
subtransient
inductances
of
the
synchronous
machine/generator and the ratio is not readily
manipulated. Another method is to manipulate the input
impedance of the converter/motor drive. This can be
accomplished by adding passive filters at the inputs,
although this can again be an expensive solution.
Herein an NSSC, Figure 12, is investigated which alters
the input impedance characteristics and adds no
additional cost. For the IMD drive i
where p denotes differentiation with respect to time.
Linearizing (1) - (2), finding the eigenvalues, and
determining the conditions for which they are in the LHP
yields the following necessary and sufficient conditions
for stability:
1)
a means of satisfying (6). However this technique is
limited because Leq and Req are both tied to the
(3)
be replaced by Tedes and T
P* <
vino 2
Req
− vino2
P*
*
inc
should
respectively.
v ( n − 1)
iinc * = in n P *
v inf
(4)
Physically speaking, (4) is normally satisfied in
practice and so (3) is the most important constraint. It is
convenient to state the stability criteria in terms of the
small signal input impedance of the constant power load.
This impedance is defined as the linearized transfer
function between the constant power load input voltage
and input current. In particular, for a constant power
load
Zin =
e,
and i
in
EFFECT OF STABILIZING CONTROL ON LOAD
IMPEDANCE - In this section the effect of the NSSC on
the load input impedance is explored. The commanded
input current of the converter can be written in terms of
the desired input current as shown in Figure 12 or in
terms of the desired input power as
and
2)
*
*
(5)
(7)
Assuming that the desired power is constant linearizing
equation (7) yields
(n − 1)vin 0
∆iinc * =
( n− 2 )
vinf 0
+
(− n)vin 0
vinf0
P*
n
( n − 1)
∆vin
(8)
P*
( n + 1)
∆vinf
From Figure 12,
∆vinf = H ( s )∆v in
This filter is designed such that H (0 ) = 1 . Therefore
(9)
In this case, and for higher powers, it can be seen that
‘n ’ acts as a gain on the filter. Although only integer
values have been considered herein, ‘n ’is in the set of
real numbers and does not have to be an integer. It is
interesting to observe that if H (s ) is set equal to
Figure 12. Nonlinear system stabilizing control
v in0 = v inf0
(10)
− 6.02 dB over the frequency range in which the
stability criteria is failing, infinite input impedance would
occur alleviating the problem. If H (s ) continued to get
smaller in magnitude the input impedance would then
become positive.
TEST SYSTEMS WITH NSSC
Incorporating (9) and (10) into (8) yields
∆iinc * =
( n − 1)P *
v in0 2
( − n )P *
∆v in +
vin0 2
H ( s )∆v in (11)
The input admittance can then be determined about the
operating point, assuming that the actual input current is
always equal to the commanded input current. In
particular,
INDUCTION
MOTOR
BASED
ELECTRIC
PROPULSION SYSTEM - The advantage of using this
simple though nonlinear stabilizing control algorithm is
that it is extremely straightforward to implement yet
highly effective in mitigating negative impedance
instabilities. In order to illustrate the effect of the
algorithm on the system, note that using the control law,
input power into the inverter is given by
n
Yinc * ( s ) =
=
∆iinc *
∆vin
(n − 1) P *
vin0
2
+
(− n) P *
vin 0
2
(12)
H (s)
v dci 

P =
~
 Pdes
v
dci


(17)
Pdes = Te, desω rm
(18)
where
Inverting the admittance yields the input impedance:
From (24) the input current may be expressed
Z inc ( s ) =
v inc 2
[n(1 − H ( s )) − 1]P *
As can be seen, the NSSC offers many possibilities for
input impedance control by adjusting ‘n ’ and H (s ) .
First, setting ‘n ’equal zero yields
−v 2
Z inc ( s ) = in0
P*
(n = 0)
(14)
whereupon it can be seen that the stabilizing control has
no effect. If ‘n ’is set equal to one, it can be seen that
with proper choice of H (s ) the input impedance can be
readily manipulated.
− v in 0 2
Z inc =
H ( s )P *
( n = 1)
(15)
(19)
Linearizing (26) about the desired operating point
( v dci = v
*
dcs )
yields
idci =
1 v *2 dcs
n − 1 Pdes
(20)
If the low pass filter time constant, τ , (used in
~ ) were so great as to not interact with
determining v
dci
the dc link dynamics and n is selected to be unity then
the input impedance presented by the inverter is infinite
for the frequency range over which negative impedance
instabilities occur, thus avoiding this type of instability.
In order to illustrate the effects of varying n and τ ,
= 400V ,
Req = 4.58 Ω , Leq = 13 .9 mH , and Ceq = 51 .4 µF .
consider the case of a system in which v
Setting ‘n ’equal to two yields
vin0 2
Zinc =
[1 − 2 H ( s )]P *
v n − 1 dci
idci = ~ n Pdes
v dci
(13)
( n = 2)
(16)
*
dceq
These parameters correspond to a test system that was
used for laboratory verification. Figure 13 illustrates the
root loci the characteristic equation as τ is varied from
0.1 ms to 1s for n =1,3,5, and 7. As can be seen, in
each case the root locus contains an unstable complex
pole (denoted A and A*) for small values of τ which
becomes stable as τ is increased. For all n shown in
Figure 13 the real part of the eigenvalues becomes more
negative as τ is increased. In addition, initially the
complex part also decreases. In the case of n = 5 ,
eventually the complex pair becomes real (point B) and
then one of these real roots meets the root
corresponding to the filter at point C, at which this pair of
eigenvalues becomes complex. In the case of n = 7
the two complex poles eventually become real at point
D, after which the pair moves away from each other on
the real axis.
change in current cannot be achieved in practice and
because the dip in link voltage causes a temporary loss
of current tracking in the hysteresis current control.
When the stabilizing controller is implemented, Figure
16, the electromagnetic torque reaches the commanded
value in the order of 8ms. Although the link stabilized
System Behavior - Incorporating the link stabilizing
control into the field oriented control is quite
straightforward. In particular, the only difference in the
control is that the instantaneous torque command is
generated using Figure 11 (with i
replaced by Te ,des , T
*
e,
*
in ,
i * inc , and vin
and v dci respectively) rather
than being set equal to the desired torque, as is
illustrated in Figure 2. The study performed on the
propulsion system earlier is repeated in Figure 14 except
that now the nonlinear stabilizing control is included.
The link stabilizing control parameters were set to n = 1
and τ = 4 ms , based on the rootlocus generated in
Figure 13. As predicted, the dc bus voltage is well
behaved and the dc link bus voltage is stable.
One concern which may arise is a possible reduction
in torque bandwidth since a drop in inverter voltage will
result in a transient dip in torque. Using a detailed
computer simulation this effect is depicted in Figures 15
and 16 with and with out the stabilizing control,
respectively. This study tests the performance of the
field oriented control to a step change in commanded
torque from 2 to 19Nm.
As can be seen, the
electromagnetic torque, in Figure 15, reaches the
commanded value in approximately 5ms. The torque
response is not instantaneous due to the fact that a step
Figure 14. Measured performance of link stabilized
field oriented control during ramp increase in desired
torque
Figure 15. Simulated performance of standard field
oriented control during step change in desired torque.
2000
1500
Imaginary Part
1000
500
Figure 16. Simulated performance of link stabilized
field oriented control during step change in desired
torque.
0
-500
-1000
-1500
-2000
-1500
-1000
-500
Real Part
0
500
Figure 13. Root Locus as τ and n are varied.
control is somewhat slower than the standard field
oriented control, this slight reduction in bandwidth is not
a significant disadvantage in view of the improved dc
bus voltage. This is particularly true due to the fact that
most propulsion systems have mechanical inertia such
that in either case the torque response may be
considered to be instantaneous.
DC POWER SYSTEM - The NSSC controller used
here sets ‘n ’equal to one leaving only the time constant
in H (s ) to be chosen, assuming a first order low pass
filter with unity gain at dc. In order to facilitate a means
of making this choice a root locus as ' τ ' was varied was
calculated by linearizing the NLAM model. This was
done with the bus filter capacitance effectively removed
from the system. The dominant poles were then plotted
in the complex plane creating a rootlocus in terms of the
filter time constant, Figure 17. From the root locus a
value of ‘τ = 2 ms ’was chosen that offered significantly
high damping but still maintained a fairly high cutoff
frequency so that system stability was guaranteed and
system performance degradation was minimized. The
resulting nonlinear PI controller with the augmented
NSSC is illustrated in Figure 18.
System Behavior – The studies presented earlier on
the DC power system are repeated in Figures 19 and 20
except that now the stabilizing controller is included. In
Figure 19 the almost identical transient response is
observed, compared to Figure 7. While in Figure 20 the
system remains stable. Notice that the generation bus
voltage does undergo increased variation; however this
is due to increased rectifier harmonics, since the source
filter capacitance has been effectively removed. The
stabilizing controller has the desired effect of maintaining
system stability for conditions (very low generation bus
capacitance) in which it was determined that a
proportional integral control alone could not maintain
500
vdc3 400
300
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
0
0
0.1
0.2
0.3
Time (sec)
0.4
0.5
400
vdc2 300
200
20
10
idcdc
Figure 19. Step in BBC load with NSSC.
stability.
CONCLUSION
2000
1500
1000
500
0
-500
-1000
-1500
-2000
-500
-400
-300
-200
-100
0
100
Figure 17. System Root Locus for Varying ‘τ’
Power systems and electric propulsion systems with
dc links are becoming more prominent in industry and
the military. This trend will continue as the need for
more efficient and versatile methods of moving energy
and designing drive systems progresses. A simplified
model of a constant power load similar to what is
typically found in many of these electric power/drive
systems was used to develop a stability criteria and a
nonlinear system stabilizing controller (NSSC).
Verification of the control was accomplished using
transient time domain studies on two such systems, a
DC power system and electric propulsion system. It was
found that the NSSC offered a means to guarantee
system stability without sacrificing significant dynamic
performance or introducing extra passive components.
In addition, the proposed strategy conveniently
separates the component regulatory aspects of the
control from the negative impedance system stability
Figure 18. Nonlinear PI controller with attached NSSC
500
v dc3
300
400
v dc2
9.
400
0
0.1
0.2
0.3
0.4
0.5
10.
300
200
20
idcdc
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
Time (sec)
0.4
0.5
11.
10
0
Figure 20. Loss of generation bus capacitance with
NSSC
aspects.
12.
13.
14.
ACKNOWLEDGMENTS
This research has been supported in part by the Naval
Sea Systems Command (Contracts N00024-93-C-4180
and N61533-95-C-0107) and P.C. Krause and
Associates' "Electric drive and finite inertia power system
analysis, modeling, simulation, and design Task 6". The
support and technical interest of Henry Hegner and
Henry Robey, the technical monitors of this project, is
gratefully acknowledged.
15.
16.
17.
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